Fitting functional response models

Functional response models usually involve non-linear functions that map prey availability to prey consumption, and such models are usually not analytic. In chapters 4 and 5, I use MCMC to fit functional response models based on the MSFR equation (2.8). The models presented in the following chapters were fitted using Fortran and R, and to verify that the fitting algorithm written in Fortran for chapter 4 was correct, its output was compared to numerical samples of

the model posterior generated by WinBUGS. In both examples, the data collection had already been completed before the modelling began, so MCMC was chosen to fit the models. But in cases where functional response data are gathered continuously, model-fitting methods such as SMC, which allow the fitting of real-time data, may be used more advantageously.

3.5.3 Uncertainty in ecological modelling

Cochrane (1999) suggests that policy-makers are reluctant to base decisions on models that fully account for all uncertainties, because output from these models is considered to be less useful than the “definite answers” of deterministic models that ignore uncertainty. However, this reluctance to take action, even when the ecological system is not understood fully, is seen as an excuse by many ecologists, who believe that management action must be taken now to avoid future damage (e. g. Dovers et al. (1996), de la Mare (1996)). In any case, the explicit modelling of uncertainties has been recognised as an important component of ecology (Ludwig

et al. (1993), Harwood and Stokes (2003)).

In multi-species contexts it is essential to state explicitly the cost-benefit functions such that different outcomes for different species may be evaluated jointly. This both makes the decision- making process more transparent and allows modellers to include all the details in a model that are relevant to the decision. When the differences between management options in terms of costs and benefits are given as probability distributions, Bayesian decision theory (e. g. Raiffa and Schlaiffer (1967) or DeGroot (1970)) can be used to choose that management decision which yields the highest expected net benefit (e. g. Wade (2000)).

In ecology, however, decision-makers may be reluctant to assign relative benefits to non-economic outcomes of their management actions (Ludwiget al. (1993), Francis and Shotton (1997)). For example, it may seem ethically wrong to associate a price tag with a probability of, say, 4% that elephants will be extinct in the wild by 2050, or simply impractical to evaluate the cost associated with such outcomes. However, even when these are not stated explicitly, managers usually follow a set of decision rules (de la Mare 1996), thereby making an implicit assessment of the trade-offs between economic and non-economic outcomes.

In particular, the precautionary principle needs to be re-evaluated in multi-species situations, because it does not extend easily to these problems (Stefansson 2003). For example, any multi- species predator-prey complex can present the problem of managing for a higher number of predators (at the risk of driving the prey population to extinction) or for a higher number of prey (by reducing the number of predators, at the risk of driving their population to extinction). The precautionary principle cannot be applied to two competing populations at once without

quantifying the relative benefits involved in this trade-off.

3.6 Summary

I have listed above the major components of uncertainty in functional response modelling (section 3.2) and given a brief overview of differences between the frequentist and Bayesian paradigms in statistics (section 3.3). Suitable numerical methods exist for fitting Bayesian functional response models (section 3.4), and I discuss how Bayesian statistics may be more suited for functional response modelling than to frequentist statistics (section 3.5). Bayesian modelling can thus help the ecological decision-making by summarising all information and their inherent uncertainty.

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4

The functional response of the hen harrier

The material presented in this section has been developed in collaboration with Sophie Smout, and under the supervision of Carmen Fern´andez, Jason Matthiopoulos and John Harwood. The presentation in this chapter is entirely my own.